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Mirrors > Home > MPE Home > Th. List > efrirr | Structured version Visualization version GIF version |
Description: A well-founded class does not belong to itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.) |
Ref | Expression |
---|---|
efrirr | ⊢ ( E Fr 𝐴 → ¬ 𝐴 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frirr 5496 | . . 3 ⊢ (( E Fr 𝐴 ∧ 𝐴 ∈ 𝐴) → ¬ 𝐴 E 𝐴) | |
2 | epelg 5431 | . . . 4 ⊢ (𝐴 ∈ 𝐴 → (𝐴 E 𝐴 ↔ 𝐴 ∈ 𝐴)) | |
3 | 2 | adantl 485 | . . 3 ⊢ (( E Fr 𝐴 ∧ 𝐴 ∈ 𝐴) → (𝐴 E 𝐴 ↔ 𝐴 ∈ 𝐴)) |
4 | 1, 3 | mtbid 327 | . 2 ⊢ (( E Fr 𝐴 ∧ 𝐴 ∈ 𝐴) → ¬ 𝐴 ∈ 𝐴) |
5 | 4 | pm2.01da 799 | 1 ⊢ ( E Fr 𝐴 → ¬ 𝐴 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2113 class class class wbr 5027 E cep 5429 Fr wfr 5475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-br 5028 df-opab 5090 df-eprel 5430 df-fr 5478 |
This theorem is referenced by: tz7.2 5503 ordirr 6184 |
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